Abstract
It is shown that Lewis’ ontological doctrine of Humean supervenience incorporates at its foundation the so-called separability principle of classical physics. In view of the systematic violation of the latter within quantum mechanics, the claim that contemporary physical science may posit non-supervenient relations beyond the spatiotemporal ones is reinforced on a foundational basis concerning constraints on the state representation of physical systems. Depending on the mode of assignment of states to quantum systems — unit state vectors versus statistical density operators — we distinguish between strongly and weakly non-Humean, non-supervenient relations. It is demonstrated that in either case, the relations of quantum entanglement constitute prototypical examples of irreducible physical relations that do not supervene upon a spatiotemporal arrangement of Humean qualities, weakening, thereby, the thesis of Humean supervenience. In this respect, the status of Lewis’ recombination principle is examined, whereas his conception of lawhood is critically investigated. It is concluded that the assumption of ontological reductionism, as expressed in Lewis’ Humean doctrine, cannot be regarded as a reliable code of the nature of the physical world and its contents. It is proposed instead that due to the undeniable existence of non-supervenient relations, a metaphysic of relations of a moderate kind ought to be acknowledged as an indispensable part of our understanding of the natural world at a fundamental level.
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Notes
There is no unanimously accepted definition either of the notion of an intrinsic property, or the conception of a qualitative property. For an introductory survey on this matter, see, Weatherson (2002), especially Section 3.1. Various suggestions for defining intrinsic properties are found, for instance, in Lewis (1986a, pp. 262–66) and, more recently, in Humberstone (1996) and Vallentyne (1997). Compare with Langton and Lewis (1998); Lewis (2001). Here, in an intuitive manner, and for purposes of fixing the terminology to be used in the sequel, a property of an object is considered as being intrinsic (and hence non-relational) if the object has this property in and of itself, independently of the existence of other objects, and a property is regarded as being qualitative if its instantiation does not depend on the existence of any particular individual. For example, the property of owing a particular thing or being the father of a particular person are common examples of non-qualitative, individual relational properties.
A possible world is unified by a natural external relation in Lewis’ sense if and only if every part of a world bears some such relation to every other part, but no part of one world ever bears any such relation to any part of another world (Lewis 1986b, Section 1.6).
Characteristic is the following claim of Hume (1975/1748, p. 74) in his “Enquiry Concerning Human Understanding”, arguing against the epistemic accessibility or, according to the customary interpretation of Hume, against the reality of necessary connections in nature:
upon the whole, there appears not, throughout all nature, any one instance of connexion which is conceivable by us. All events seem entirely loose and separate. One event follows another; but we never can observe any tie between them. They seem conjoined, but never connected. And as we can have no idea of any thing which never appeared to our outward sense or inward sentiment, the necessary conclusion seems to be that we have no idea of connexion or power at all, and that these words are absolutely without any meaning, when employed either in philosophical reasonings or common life.
Akin to both principles of separability and Humean supervenience is the thesis of ‘local physicalism’ or ‘particularism’ as has been variously called by Teller (1986, 1989) in order to encapsulate the mechanistic worldview of classical physics. According to Teller (1989, p. 213), particularism holds that the basic inhabitants of our world are distinct individuals, possessing non-relational properties, and that all relations between individuals supervene on the non-relational properties of the relata. This means that if a, b, c,… is a set of individual objects, then, any relational properties and physical relations holding among the relata a, b, c,… supervene upon their non-relational properties. It should be underlined that although the doctrines of Humean supervenience and particularism bear strong similarities to each other, they are not conceptually identical. It is not clear, for instance, whether, according to Teller’s notion of particularism, spatial or spatiotemporal relations enjoy a privileged status. If particularists treat all relations alike, then spatiotemporal relations ought to supervene upon the non-relational properties of space-time points or of point-objects that occupy space-time points. Hence, in such a circumstance, there appears a difference with the thesis of Humean supervenience, according to which, spatiotemporal relations belong to the subvenient basis, namely, to the set of basic lower-level properties on which upper-level properties supervene. Furthermore, the range of application of each doctrine varies. In contrast to the conception of Humean supervenience, which concerns all the properties and relations that do not belong to the subvenient basis, particularism concerns only the relations holding among individuals.
It is of no coincidence, in this respect, that Lewis’ examples of ‘Humean’ properties, namely, intrinsic properties that require no more than a spatiotemporal point to be instantiated, are the values of the electromagnetic and gravitational fields (see also Loewer 1996, p. 102). On the other hand, Robinson (1989) and, more recently, Oppy (2000) and Butterfield (2006) have argued that vectorial (or, more generally, tensorial) quantities cannot be ‘Humean’ properties in Lewis’ sense, because vector-valued magnitudes cannot be instantiated at ‘isolated’ points, as no direction can be associated with ‘isolated’ points. We need the notion of the ‘neighbourhood’ of a point in order to define a tangent vector and hence a specific direction. Admittedly, this necessity is explicit in the case of the coordinate-dependent definition of a tangent vector at a point, where in order to define the coordinates of the vector, we necessarily use coordinate variations (differentials) in an infinitesimal area around the point. Even in the case of coordinate-independent definitions — the vector being considered as an equivalence class of curves on the manifold, or as a differential operator acting on differentiable functions of the manifold — there is an implicit necessity in using the notion of the neighbourhood of a point. This being so, we take nonetheless the view that it is not necessary to put much weight on the notion of ‘neighbourhood’ while investigating the issue of Humean supervenience and its philosophical implications. Thus, from now on, we shall take for granted that tensorial quantities instantiated at space–time points are the most plausible candidates for the properties of the Humean subvenient basis.
Two points of clarification should be noted: First, the symbols “□” and “◇” denote the modal operators for necessity and possibility. Thus, “□P” should be read “it is necessary that P” and “◇ P” should be read “it is possible that P”. Second, the differentiation between ‘determinable’ and ‘determinate’ should be understood in the sense of a determinable attribute of a certain kind — such as having spin — and a determinate attribute — such as having spin of, say, 1/2 along a certain direction.
In this work, we shall not consider in any detail alternative interpretations to Hilbert-space quantum mechanics as, for instance, Bohm’s ontological or causal interpretation.
Lewis touches upon the concept of quantum states, especially the superposition principle of states in quantum mechanics, in the introductory section of his lecture “How Many Lives Has Schrödinger’s Cat?”, published three years after his untimely death in 2001. Lewis seems to understand a superposed state as a complex of concomitant layers of reality, as a plurality of coexisting actualities. Thus, by referring to the example of the benzene ring, he writes: “….a molecule with an objectively indeterminate structure [such as the benzene ring] is really two coexisting molecules, one with one structure and one with the other (or at any rate, two things that are molecule-like…)” (Lewis 2004, p. 4). Lewis’ rather short account of the quantum superposition concept is sketchy in certain respects, as he himself admits, whereas his conception of ‘coexisting actualities’ with respect to a superposed state certainly seems ontologically extravagant; instead of one molecule we get a superimposition of probably infinitely many different molecule-like things. This need not be very repugnant for Lewis, in view of his willingness to embrace a kindred doctrine concerning the reality of possible worlds, a doctrine that harks back in his commitment to the actual existence of all those alternative ‘realities’ entailed by forms of modal reasoning. When it comes to Hilbert-space quantum mechanics, however, we think Lewis’ account of the quantum superposition concept is flawed. According to standard interpretational ideas, quantum mechanical superpositions have an interpretation not in terms of what is actual, but rather of what is potentially realizable via a probability distribution of possible values for the various physical quantities of a system out of which only one is actualized when the system is interacting with its environment or a pertinent experimental context. Consequently, a superposed state or a general quantum mechanical pure state may be construed in an ontic sense, regardless of any operational procedures, as representing a network of potentialities, namely, an intertwined set of potentially possible and not actually coexisting events (cf. e.g. Margenau 1950, pp. 335–37, 452–54; Heisenberg 1974, pp. 16–17; Shimony 1993, chapter 11 ; Karakostas 2007, pp. 284–287). In the case of the spin-singlet state W S of Eq. (2), for example, no definite spin state can be assigned to either of the particles S1 and S2, since neither of the corresponding unit vectors \( \left| {\psi _{ \pm } > _{1} } \right. \) and \(\left| {\phi _ \pm >_2 } \right.\) are eigenstates of W S; the state W S is an eigenstate of the total spin operator \(\sigma _{{\text{1}}} \otimes I + I \otimes \sigma _{2} \), which cannot be understood as being composed of definite individual spin values of the two single particles. Hence, in the state W S, no spin component of either particle S1 or particle S2 exists in an actual form, possessing occurrent spin properties. All three spin components of each particle, however, coexist in a potential form and any one component possesses the tendency of being actualized at the expense of the indiscriminacy of the others if the associated particle interacts with an appropriate measuring apparatus. In this respect, the spin-singlet state — as any compound superposed state — represents in essence the entanglement, the inseparable correlation of potentialities, whose content is not exhausted by a catalogue of actual preexisting values that may be assigned to the spin properties of S1 and S2, separately. As explained in the text, this is an instance, altogether alien to Lewis’ ideas, of the peculiar ‘holism’ of quantum mechanics: definite properties of a compound system do not supervene on properties of its parts and their spatiotemporal relations.
It is well known that spin-singlet correlations violate Bell’s inequalities. We note in this connection the interesting result of Gisin (1991), Popescu and Rohrlich (1992) that for any entangled state of a two-component system there is a proper choice of pairs of observables whose correlations do violate Bell’s inequality.
As an attempt to rescue Humean supervenience from the challenge that quantum non-separability poses, Loewer (1996, p. 104) has suggested that as the ‘‘fundamental space of the world’’ be taken to be abstract configuration space, rather than ordinary space-time. In this view — and within the context of Bohm’s theory, or more appropriately, a version of it — the quantum state is construed as a genuine field in configuration space, where a three-dimensional arrangement of an N-particle system corresponds to a single point in 3N dimensions. This, if the property of spin is neglected, otherwise the dimensionality of configuration space increases. The value of the field at any single point in configuration space, then, represents the amplitude of the quantum state at that point. The hope for Humean supervenience is that although fundamental properties are not instantiated at single points or arbitrarily small regions of space–time, once we realize that the ‘‘actual’’ or ‘‘fundamental space of the world’’ is 3N-dimensional configuration space those properties turn out to be ‘local’ after all. However, it is not clear whether Lewis’ thesis of Humean supervenience retains its substantial content after the move to highly abstract configuration space. The latter is not the concrete spatiotemporal continuum aspired by Lewis. To the extent that the thesis of Humean supervenience enjoys intuitive support, its source no doubt arises from considerations in quality arrangement in space-time. Secondly, even supposing that all presuppositions of Loewer’s suggestion do hold, there is no sufficient reason to demand that qualities distributed in configuration space ought to be intrinsic to the points of that space. Thirdly, any considerations about the ‘‘fundamental space of the world’’ should be grounded in physical science, not in a priori philosophical assumptions. In this respect, contemporary developments in theoretical physics strongly suggest that space-time itself is an approximate, derived notion (e.g., Witten 1996; Butterfield and Isham 2001). On such an account, spatiotemporal relations are no longer fundamental. Hence, the letter of Lewis’ formulation of Humean supervenience will be violated. Nonetheless, we shall take no consideration of these possibilities, for even confining ourselves to existing mature, experimentally confirmed physical theories, there exist adequate ground to resist the thesis that Humean supervenience holds in our world.
It is worthy to note that the non-purity of the subsystem states of Eq. 4 arises as a restriction of the overall pure state WS of the entangled system to the observables pertaining to a component subsystem. Any subsystem state in this situation is exclusively defined at the level of the whole system; there is no individual state for a component subsystem alone. For, there exists no justification in regarding the prescribed reduced states of Eq. 4 as being associated with any specific ensemble of pure states with corresponding eigenvalue probabilities. In this respect, the reference of a reduced state is only of a statistical, epistemic nature. It simply reflects the statistics that may be derived by a series of local measurements performed on a given component subsystem.
Hughston et al. (1993) provide a constructive classification of all discrete ensembles of pure quantum states that correspond to a fixed density operator.
In Lewis’ philosophical scheme, the notion of causation should be understood in a broadly neo-Humean way as a contingent relation between ‘distinct existences’. Causal relationships are, roughly speaking, nothing but patterns which supervene on a point-by-point distribution of properties.
It should be noted that the so-called invariant or state-independent properties — like ‘rest-mass’, ‘charge’ and ‘spin’ — of elementary objects–systems can only characterize a certain class of objects; they can only specify a certain sort of particles, e.g., electrons, protons, neutrons, etc. They are not sufficient, however, for determining a member of the class as an individual object, distinct from other members within the same class, that is, from other objects having the same state-independent properties. Thus, an ‘electron’, for instance, could not be of the particle-kind of ‘electrons’ without fixed, state-independent properties of ‘mass’ and ‘charge’, but these in no way suffice for distinguishing it from other similar particles or for ‘individuating’ it in any particular physical situation. For a detailed treatment of this point, see, for example, French and Krause (2006); Castellani (1999).
The seemingly tempting thought of including into the supervenience basis the quantum entangled relations among the related parts of a quantum whole would certainly be inappropriate. For, firstly, such a manoeuvre would run against the very essence of Lewis’ thesis of Humean supervenience, envisaging a world of unconnected, independently existing particulars. And, secondly, in such a circumstance any relevant supervenience claim would be rendered trivial. It is essential that in specifying a supervenience basis one should avoid adding global properties or relations to this basis. The latter is expected to include, according to Healey (1991, p. 401), only “the qualitative, intrinsic properties and relations of the parts, i.e., the properties and relations that these bear in and of themselves, without regard to any other objects, and irrespective of any further consequences of their bearing these properties for the properties of any wholes they might compose”.
The qualification concerns models characterized by ‘extremely’, according to the present state of physical science, counter-intuitive features.
It is worth noting that the thesis of Humean supervenience is not identical to physicalism or physicalistic reductionism. It is rather a particularly strong version of it. For, one need not uphold Humean supervenience in order to support a robust physicalist doctrine, in view of which, all contingent matters of fact supervene on physical matters of fact.
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Acknowledgements
For discussion and comments on previous versions, I thank participants at audiences in the Fifth European Conference of Analytic Philosophy (Lisbon) and Thirteenth International Congress of Logic, Methodology and Philosophy of Science (Beijing).
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Karakostas, V. Humean Supervenience in the Light of Contemporary Science. Int Ontology Metaphysics 10, 1–26 (2009). https://doi.org/10.1007/s12133-008-0037-8
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DOI: https://doi.org/10.1007/s12133-008-0037-8